#7560483

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,17)(15,19,22,26)(16,20,25,24)(21,27,28,23);(2,3)(6,9)(10,12)(13,15,18,22)(14,19,17,26)(16,23,25,27)(20,21,24,28);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,14,18,17)(15,19,22,26)(16,24,25,20)(21,23,28,27);(1,3,4,2)(13,16)(14,20)(15,21)(17,24)(18,25)(19,27)(22,28)(23,26);(6,9)(10,12)(16,25)(20,24)(21,28)(23,27)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467679);